#7727974

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

5672 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)(3,7)(4,5)(6,8)(9,10,13,15)(11,16,14,12);(9,11)(10,12)(13,14)(15,16)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32);(2,5)(6,7)(9,12,13,16)(10,14,15,11)(18,23)(20,24)(26,29)(28,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,14)(15,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27);(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261948, #72262090, #72262220
maximal normal subgroup of	#72261948, #72262090, #72262220

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7635170);