#383971

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291167);