#411732

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1952 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)(7,11)(8,18)(9,21)(10,13)(12,19)(14,22)(15,27)(16,23)(17,31)(20,32)(24,29)(25,30)(26,33)(28,35)(34,36);(11,23)(24,34)(25,31)(27,35);(2,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,24)(13,23)(14,25)(18,27)(19,29)(20,28)(21,31)(22,30)(26,34)(32,35)(33,36);(5,8,9,19,10,20,22,33)(6,12,14,18,13,26,21,32)(7,15,17,29,16,28,30,36)(11,24,25,27,23,34,31,35)

Polycyclic presentation:

2^9|d^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231499, #72231524, #72231526, …
maximal normal subgroup of	#72231499, #72231524, #72231526, …
maximal quotient of	#72314528, #72314529

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318928);