#8508957

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312133
maximal normal subgroup of	#72312133

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416153);