#7626896

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3515	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(1,2,4,3)(7,13)(11,14)(15,19)(18,20)(21,22,26,27)(23,28,30,25)(24,32,34,29)(31,33,36,35);(5,6,10,8)(7,11,13,14)(9,12,17,16)(15,18,19,20)(22,27)(23,30)(29,32)(31,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,23)(22,28)(24,31)(25,27)(26,30)(29,35)(32,33)(34,36);(5,8,10,6)(7,14,13,11)(9,12,17,16)(15,18,19,20)(21,24,26,34)(22,29,27,32)(23,31,30,36)(25,33,28,35);(1,3,4,2)(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,25,26,28)(22,23,27,30)(24,33,34,35)(29,31,32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534092);