#421696

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

842 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)(5,6,10,8,11,14,16,22)(7,15,18,28,19,20,26,12)(9,13,24,21,25,30,31,34)(17,29,23,35,32,33,36,27);(5,7,16,18,11,19,10,26)(6,12,22,15,14,28,8,20)(9,23,31,32,25,36,24,17)(13,29,34,35,30,33,21,27);(5,8,10,14,11,22,16,6)(7,15,18,28,19,20,26,12)(9,21,24,30,25,34,31,13)(17,29,23,35,32,33,36,27);(2,4)(5,9,10,24,11,25,16,31)(6,13,8,21,14,30,22,34)(7,17,18,23,19,32,26,36)(12,27,15,29,28,35,20,33)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322778
maximal normal subgroup of	#72322778

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328892);