#7626137

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator	C₂²	C₂⁵×C₄
Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3482	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6688 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)(5,6)(7,15)(8,13)(9,14)(10,12)(11,16)(17,20)(18,19)(21,22)(23,29)(24,32)(25,27)(26,30)(28,33)(31,34)(35,36);(5,7)(6,11)(8,17)(9,18)(10,16)(12,15)(13,19)(14,20)(21,23)(22,26)(24,31)(25,30)(27,29)(28,34)(32,35)(33,36);(7,16)(11,15)(17,18)(19,20)(22,27)(23,30)(28,32)(31,36);(1,3,4,2)(6,12)(11,15)(13,14)(19,20)(22,27)(23,30)(28,32)(31,36);(5,8,10,9)(6,13,12,14)(7,17,16,18)(11,19,15,20)(21,24,25,33)(22,28,27,32)(23,31,30,36)(26,34,29,35);(5,9,10,8)(6,14,12,13)(7,18,16,17)(11,20,15,19)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533333);