#544311

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

2040 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gh,e^2=f,b^a=bf,c^a=cg,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 451507);