#7626810

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×#3459	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6980 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)(7,12)(8,18)(9,19)(10,13)(11,17)(14,20)(15,16)(21,22,28,31)(23,29,33,36)(24,34,26,30)(25,27,35,32);(5,7)(6,11)(8,15)(9,16)(10,17)(12,13)(14,19)(18,20)(21,23,28,33)(22,29,31,36)(24,32,26,27)(25,30,35,34);(1,3,4,2)(5,8)(6,9)(7,14)(10,19)(11,20)(12,16)(13,18)(15,17)(21,24,28,26)(22,30,31,34)(23,32,33,27)(25,29,35,36);(5,8)(6,9)(7,15)(10,19)(11,16)(12,20)(13,18)(14,17)(21,25)(22,27)(23,30)(24,29)(26,36)(28,35)(31,32)(33,34);(5,7)(6,12)(8,14)(9,16)(10,17)(11,13)(15,19)(18,20)(21,26,28,24)(22,30,31,34)(23,27,33,32)(25,29,35,36);(5,9)(6,8)(7,16)(10,18)(11,15)(12,14)(13,19)(17,20)(21,27)(22,25)(23,24)(26,33)(28,32)(29,30)(31,35)(34,36)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,e^2=gh,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, 7534006);