#7908670

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

5221 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318899, #72319623, #72319686
maximal normal subgroup of	#72318899, #72319623, #72319686

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7815866);