#400549

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

Polycyclic presentation:

2^9|a^2=f,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	#72320778
maximal normal subgroup of	#72320778

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307745);