#422300

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#2010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1210 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)(7,15,18,29)(8,12)(9,13)(10,14)(11,17,24,22)(16,26,31,35)(19,32,33,25)(20,27)(21,28)(23,30)(34,36);(5,7,9,22,10,18,23,17)(6,11,13,29,14,24,30,15)(8,19,21,16,20,33,34,31)(12,26,28,25,27,35,36,32);(8,20)(11,24)(12,27)(15,29)(16,31)(19,33)(21,34)(28,36);(2,4)(5,8,9,21,10,20,23,34)(6,12,13,28,14,27,30,36)(7,16,17,19,18,31,22,33)(11,25,15,26,24,32,29,35)

Polycyclic presentation:

2^9|a^2=f,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, 329496);