#7823090

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

5864 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)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32);(9,11)(10,12)(13,14)(15,16)(17,19,22,26)(18,23,25,30)(20,27,24,31)(21,28,29,32);(2,5)(6,7)(9,12,13,16)(10,14,15,11)(19,26)(23,30)(27,31)(28,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,14)(15,16)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30);(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261968
maximal normal subgroup of	#72261968

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7730286);