#7702855

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4820 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)(4,6)(5,8)(9,10,13,15)(11,16,12,14)(17,18)(19,25)(20,28)(21,23)(22,26)(24,30)(27,31)(29,32);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,22,21)(18,23,26,25)(20,27,30,29)(24,31,28,32);(10,15)(11,12)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,19,22,21)(18,25,26,23)(20,29,30,27)(24,31,28,32);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,16,15,14)(17,21,22,19)(18,25,26,23)(20,29,30,27)(24,32,28,31);(1,4,5,3)(2,6,8,7)

Polycyclic presentation:

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

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, 7610051);