#360199

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1578 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)(5,6,10,14,11,15,9,25)(7,16,20,26,19,29,18,12)(8,21,24,28,22,35,23,13)(17,31,34,36,32,27,33,30);(1,2,4,3)(5,7,9,18,11,19,10,20)(6,12,14,16,15,26,25,29)(8,17,23,33,22,32,24,34)(13,27,21,30,28,31,35,36);(8,22)(12,26)(13,28)(16,29)(17,32)(21,35)(23,24)(33,34);(1,3,4,2)(5,8,9,23,11,22,10,24)(6,13,14,21,15,28,25,35)(7,17,18,33,19,32,20,34)(12,27,16,30,26,31,29,36)

Polycyclic presentation:

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

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, 267395);