#401231

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

975 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)(5,6,11,16)(7,17)(8,23,26,13)(9,14,27,25)(10,15,18,29)(12,28)(19,20)(21,32)(22,33)(24,36)(30,31)(34,35);(5,7,18,21,11,22,10,28)(6,12,29,17,16,32,15,33)(8,24,14,31,26,19,25,34)(9,20,23,35,27,36,13,30);(5,8,10,25,11,26,18,14)(6,13,15,27,16,23,29,9)(7,19,21,34,22,24,28,31)(12,30,17,20,32,35,33,36);(2,4)(5,9,18,23,11,27,10,13)(6,14,29,26,16,25,15,8)(7,20,21,35,22,36,28,30)(12,31,17,19,32,34,33,24)

Polycyclic presentation:

2^9|a^2=f,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	#72322108
maximal normal subgroup of	#72322108

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308427);