#377079

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=eh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,d^c=di,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, 284275);