#360219

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	#2487
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1566 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=e,e^2=i,f^2=i,h^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,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249978, #72249991, #72250000, …
maximal normal subgroup of	#72249978, #72249991, #72250000, …
maximal quotient of	#72265355

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267415);