#371088

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

872 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250386, #72252977, #72256881
maximal normal subgroup of	#72250386, #72252977, #72256881

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278284);