#364106

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1190 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)(5,6,12,18,10,16,11,17)(7,19,9,15,21,13,22,28)(8,14,24,31,25,32,26,33)(20,27,23,29,35,34,36,30);(5,7,11,22,10,21,12,9)(6,13,17,15,16,19,18,28)(8,23,24,35,25,36,26,20)(14,29,31,34,32,30,33,27);(5,8)(6,14)(7,20)(9,23)(10,25)(11,26)(12,24)(13,27)(15,30)(16,32)(17,33)(18,31)(19,34)(21,35)(22,36)(28,29);(5,9,12,21,10,22,11,7)(6,15,18,13,16,28,17,19)(8,23,24,35,25,36,26,20)(14,30,31,27,32,29,33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239720, #72239887, #72250581, …
maximal normal subgroup of	#72239720, #72239887, #72250581, …
maximal quotient of	#72239727

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271302);