#248830

order	512
nilpotency class	3
isoclinism	isoclinic class of #248743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61707	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,d^c=di,e^c=eh,e^d=eh,f^d=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156026);