#7673721

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5598 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342303, #72342320, #72342329, …
maximal normal subgroup of	#72342303, #72342320, #72342329, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7580917);