#7674709

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
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6368 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)(3,7)(4,5)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,22)(20,27)(21,23)(24,28)(25,30)(26,32)(29,31);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,14,15,12)(17,19,21,25)(18,22,23,30)(20,26,28,31)(24,29,27,32);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,14)(15,16)(18,23)(19,25)(24,27)(26,31);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,22)(20,24)(21,23)(25,30)(26,29)(27,28)(31,32);(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31);(17,20)(18,24)(19,26)(21,28)(22,29)(23,27)(25,31)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581905);