#400946

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ 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)(7,15,18,27,19,20,25,12)(9,13,23,21,24,28,30,34)(17,29,31,36,32,33,35,26);(5,7,16,18,11,19,10,25)(6,12,22,15,14,27,8,20)(9,17,30,31,24,32,23,35)(13,26,34,29,28,36,21,33);(5,8,10,14,11,22,16,6)(7,15,18,27,19,20,25,12)(9,21,23,28,24,34,30,13)(17,29,31,36,32,33,35,26);(2,4)(5,9,16,30,11,24,10,23)(6,13,22,34,14,28,8,21)(7,17,18,31,19,32,25,35)(12,26,15,29,27,36,20,33)

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320982
maximal normal subgroup of	#72320982

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308142);