#6376613

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34880	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=cf,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283809);