#7626889

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×#3520	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6824 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)(7,14)(8,17)(9,12)(10,16)(11,13)(15,19)(18,20)(23,29)(27,30)(31,35)(34,36);(5,7)(6,10)(8,18)(9,15)(11,16)(12,20)(13,14)(17,19)(21,22,26,24)(23,27,29,30)(25,28,33,32)(31,34,35,36);(5,8,11,9)(6,12,13,17)(7,15,16,18)(10,19,14,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,7)(6,10)(8,15)(9,18)(11,16)(12,19)(13,14)(17,20)(21,24,26,22)(23,30,29,27)(25,28,33,32)(31,34,35,36);(1,3,4,2)(5,9,11,8)(6,12,13,17)(7,15,16,18)(10,20,14,19)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36);(5,10)(6,7)(8,19)(9,20)(11,14)(12,15)(13,16)(17,18)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gh,e^2=gi,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, 7534085);