#7672664

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#35256	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5246 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579860);