#7626819

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	C₂⁵	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=gh,e^2=h,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, 7534015);