#429992

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2404
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1619 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)(7,15,19,24)(8,12)(9,13)(10,14)(11,18,25,17)(16,31,33,30)(20,27)(21,28)(22,29)(23,34,26,32)(35,36);(5,7,9,17,10,19,22,18)(6,11,13,24,14,25,29,15)(8,16,21,34,20,33,35,32)(12,26,28,30,27,23,36,31);(5,8)(6,12)(7,16)(9,21)(10,20)(11,23)(13,28)(14,27)(15,30)(17,34)(18,32)(19,33)(22,35)(24,31)(25,26)(29,36);(2,4)(7,17,19,18)(8,20)(11,24,25,15)(12,27)(16,32,33,34)(21,35)(23,30,26,31)(28,36)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=f,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.

Relationships

maximal subgroup of	#72285534, #72285552, #72285668, …
maximal normal subgroup of	#72285534, #72285552, #72285668, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337188);