#258137

order	512
nilpotency class	3
isoclinism	isoclinic class of #246079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25)(19,26,29,23)(20,30,22,24)(27,32,28,31);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,26)(30,32);(13,16)(19,27)(20,22)(23,31)(24,30)(26,32)(28,29);(1,3,4,8)(2,6,5,7)(17,20,21,22)(18,24,25,30)(19,28,29,27)(23,31,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309297, #72309299, #72309304
maximal normal subgroup of	#72309297, #72309299, #72309304

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 165333);