#7918140

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

4924 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=g,e^2=g,f^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	#72256737, #72316538
maximal normal subgroup of	#72256737, #72316538
maximal quotient of	#72256708

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7825336);