#7626835

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×#3455	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6624 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)(5,6,12,10)(7,13,18,17)(8,19,9,14)(11,20,16,15)(21,22)(23,32)(24,33)(25,26)(27,34)(28,31)(29,36)(30,35);(5,7,12,18)(6,13,10,17)(8,16,9,11)(14,20,19,15)(21,23,28,26)(22,25,31,32)(24,34,36,30)(27,33,35,29);(1,3,4,2)(5,8,12,9)(6,14,10,19)(7,16,18,11)(13,20,17,15)(21,24)(22,29)(23,30)(25,35)(26,34)(27,32)(28,36)(31,33);(5,9,12,8)(6,14,10,19)(7,11,18,16)(13,20,17,15)(21,25,28,32)(22,23,31,26)(24,35,36,27)(29,30,33,34);(5,10,12,6)(7,17,18,13)(8,19,9,14)(11,20,16,15)(21,26,28,23)(22,25,31,32)(24,34,36,30)(27,29,35,33);(5,11)(6,15)(7,8)(9,18)(10,20)(12,16)(13,14)(17,19)(21,27,28,35)(22,30,31,34)(23,33,26,29)(24,32,36,25)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=h,e^2=h,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, 7534031);