#7744968

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

5029 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7652164);