#97249

order	512
nilpotency class	3
isoclinism	isoclinic class of #97214
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

641 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 641 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,7,8,5)(4,6)(9,10,14,20,15,19,13,18)(11,21,24,16,25,32,23,31)(12,26,27,34,28,17,29,33)(22,36,37,30,38,35,39,40)(41,42)(43,51,54,64)(44,55,57,48)(45,49)(46,50)(47,53,62,58)(52,67)(56,65,59,63)(60,66)(61,68)(69,72)(70,71);(1,3,4,8)(2,5,6,7)(9,11,15,25)(10,16,19,31)(12,22,28,38)(13,24,14,23)(17,30,26,40)(18,32,20,21)(27,39,29,37)(33,36,34,35)(41,43,46,54)(42,47,50,62)(44,52,57,69)(45,58,60,53)(48,61,55,71)(49,64,66,51)(56,70,59,68)(63,67,65,72);(9,12,15,28)(10,17,19,26)(11,22,25,38)(13,27,14,29)(16,30,31,40)(18,33,20,34)(21,35,32,36)(23,37,24,39)(41,44,46,57)(42,48,50,55)(43,52,54,69)(45,59,60,56)(47,61,62,71)(49,65,66,63)(51,67,64,72)(53,70,58,68)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=h,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=ei,f^b=fgh,d^c=di,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4445);