#7715780

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4624 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)(3,4)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,27,28,24)(21,25,31,29)(22,30,32,26);(9,11,13,16)(10,14,12,15)(17,19,23,28)(18,24,20,27)(21,30,31,26)(22,29,32,25);(2,6)(3,7)(19,28)(21,31)(24,27)(25,29);(1,3,5,7)(2,4,6,8)(9,10,13,12)(11,14,16,15)(17,20,23,18)(19,27,28,24)(21,29,31,25)(22,30,32,26);(1,4)(2,7)(3,6)(5,8)(9,12,13,10)(11,15,16,14)(17,21,23,31)(18,25,20,29)(19,26,28,30)(22,24,32,27);(17,22)(18,26)(19,29)(20,30)(21,24)(23,32)(25,28)(27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622976);