#360196

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
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	#2408
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1674 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)(7,15)(8,19)(9,22)(10,14)(11,17)(12,20)(13,23)(16,29)(18,27)(21,33)(24,32)(25,31)(26,34)(28,30)(35,36);(5,7,9,17,10,18,23,32)(6,11,13,15,14,24,22,27)(8,16,20,30,21,31,34,36)(12,25,19,28,26,29,33,35);(11,24)(15,27)(25,29)(28,35);(5,8,9,20,10,21,23,34)(6,12,13,19,14,26,22,33)(7,16,17,30,18,31,32,36)(11,25,15,28,24,29,27,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265371, #72265390, #72265395, …
maximal normal subgroup of	#72265371, #72265390, #72265395, …
maximal quotient of	#72313849

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267392);