#367991

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=e,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275187);