#376873

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	#23406	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1466 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)(5,6,10,14,11,15,9,23)(7,12,19,24,18,26,16,27)(8,13,22,29,20,28,21,33)(17,25,31,34,32,36,30,35);(2,4)(5,7)(6,12)(8,17)(9,16)(10,19)(11,18)(13,25)(14,24)(15,26)(20,32)(21,30)(22,31)(23,27)(28,36)(29,34)(33,35);(2,4)(7,16,18,19)(8,20)(12,24,26,27)(13,28)(17,31,32,30)(21,22)(25,35,36,34)(29,33);(2,4)(5,8,10,22,11,20,9,21)(6,13,14,29,15,28,23,33)(7,17,19,31,18,32,16,30)(12,25,24,34,26,36,27,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249969, #72252980, #72256886
maximal normal subgroup of	#72249969, #72252980, #72256886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284069);