#450121

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1050 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357317);