#358642

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

778 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:

(5,6)(7,17,21,27)(8,22)(9,13)(10,14)(11,16)(12,20,28,34)(15,25)(18,26,32,36)(19,31,33,35)(23,29)(24,30);(1,2)(3,4)(5,7,10,20,11,21,25,34)(6,12,15,27,16,28,14,17)(8,18,9,19,23,32,24,33)(13,26,22,35,30,36,29,31);(2,4)(5,8,10,9,11,23,25,24)(6,13,15,22,16,30,14,29)(7,18,20,19,21,32,34,33)(12,26,27,35,28,36,17,31);(5,9,10,23,11,24,25,8)(6,13,14,29,16,30,15,22)(7,19,20,32,21,33,34,18)(12,26,17,31,28,36,27,35)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,h^b=hi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240851, #72252972, #72256876
maximal normal subgroup of	#72240851, #72252972, #72256876

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265838);