#400261

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

Polycyclic presentation:

2^9|a^2=e,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	#72321017
maximal normal subgroup of	#72321017

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307457);