#6393697

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#3482	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3111 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)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,26,27,25,30,23,29)(24,28,32,34,33,36,31,35);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,25,26)(22,27,30,29)(24,32,33,31)(28,35,36,34);(1,2,3,4)(21,24,26,32,25,33,23,31)(22,28,27,34,30,36,29,35);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,23,25,26)(22,29,30,27)(24,31,33,32)(28,35,36,34);(6,12)(10,13)(11,20)(18,19)(22,30)(27,29)(28,36)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi

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, 6300893);