#112159

order	512
nilpotency class	3
isoclinism	isoclinic class of #112156
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

731 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 731 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=dhi,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, 19355);