#426461

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

801 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)(5,6,10,15,11,8,12,23)(7,16,18,21,20,31,19,13)(9,14,25,30,26,22,27,36)(17,29,24,34,33,35,32,28);(5,7,12,18,11,20,10,19)(6,13,23,16,8,21,15,31)(9,24,27,33,26,32,25,17)(14,29,36,34,22,35,30,28);(5,8,12,15,11,6,10,23)(7,16,18,21,20,31,19,13)(9,22,27,30,26,14,25,36)(17,29,24,34,33,35,32,28);(2,4)(5,9,10,25,11,26,12,27)(6,14,15,30,8,22,23,36)(7,17,18,24,20,33,19,32)(13,28,16,29,21,34,31,35)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=ef,d^2=e,e^2=i,g^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,g^b=gi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322821
maximal normal subgroup of	#72322821

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333657);