#7674700

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#43489	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4846 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581896);