#371154

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

1122 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)(5,6,11,17,12,18,10,16)(7,13,21,15,20,29,9,26)(8,14,25,32,24,31,27,33)(19,34,22,36,35,28,23,30);(1,2,4,3)(5,7,11,21,12,20,10,9)(6,13,17,15,18,29,16,26)(8,22,27,19,24,23,25,35)(14,30,33,28,31,36,32,34);(5,8)(6,14)(7,19)(9,23)(10,27)(11,25)(12,24)(13,28)(15,30)(16,33)(17,32)(18,31)(20,35)(21,22)(26,36)(29,34);(1,3,4,2)(5,9,10,20,12,21,11,7)(6,15,16,13,18,26,17,29)(8,23,27,35,24,22,25,19)(14,30,33,28,31,36,32,34)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,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	#72252681, #72256873, #72256881
maximal normal subgroup of	#72252681, #72256873, #72256881

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278350);