#254528

order	512
nilpotency class	3
isoclinism	isoclinic class of #254515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1543 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,5)(4,6)(7,8)(9,10,14,18)(11,15,21,30)(12,22,23,16)(13,25,27,17)(19,32,26,28)(20,29,34,35)(24,31,37,36)(33,40,38,39);(1,3)(2,5)(4,7)(6,8)(11,19,21,26)(15,28,30,32)(20,33,34,38)(29,39,35,40);(5,8)(9,11)(10,15)(12,20)(13,26)(14,21)(16,29)(17,32)(18,30)(19,27)(22,35)(23,34)(24,38)(25,28)(31,40)(33,37)(36,39);(9,12,14,23)(10,16,18,22)(11,20,21,34)(13,24,27,37)(15,29,30,35)(17,31,25,36)(19,33,26,38)(28,39,32,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,g^c=gh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255929, #72256151, #72256161
maximal normal subgroup of	#72255929, #72256151, #72256161

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161724);