#330297

order	512
nilpotency class	3
isoclinism	isoclinic class of #329647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

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, 237493);