#9815056

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

5362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336552
maximal normal subgroup of	#72336552
maximal quotient of	#72336660

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9722252);