#376903

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	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1478 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)(5,6,11,17,10,15,12,16)(7,18,8,21,20,13,23,30)(9,24,27,32,26,14,28,33)(19,29,22,36,34,31,25,35);(2,4)(5,7)(6,13)(8,11)(9,19)(10,20)(12,23)(14,29)(15,18)(16,21)(17,30)(22,27)(24,31)(25,28)(26,34)(32,35)(33,36);(2,4)(5,8,12,7,10,23,11,20)(6,13,17,30,15,18,16,21)(9,25,28,34,26,22,27,19)(14,31,33,35,24,29,32,36);(2,4)(5,9,11,27,10,26,12,28)(6,14,17,33,15,24,16,32)(7,19,8,22,20,34,23,25)(13,29,30,36,18,31,21,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252634, #72256873, #72256886
maximal normal subgroup of	#72252634, #72256873, #72256886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284099);