#379480

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

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

734 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)(5,6,11,17,12,18,10,16)(7,13,9,25,20,28,21,15)(8,14,24,32,23,31,26,33)(19,34,22,30,35,27,36,29);(5,7,10,21,12,20,11,9)(6,13,16,15,18,28,17,25)(8,22,24,35,23,36,26,19)(14,29,32,34,31,30,33,27);(5,8,10,26,12,23,11,24)(6,14,16,33,18,31,17,32)(7,19,21,36,20,35,9,22)(13,27,15,30,28,34,25,29);(1,3,4,2)(5,9,11,20,12,21,10,7)(6,15,17,13,18,25,16,28)(8,22,24,35,23,36,26,19)(14,30,32,27,31,29,33,34)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,d^2=h,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	#72265431, #72307302, #72307337
maximal normal subgroup of	#72265431, #72307302, #72307337

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286676);