#495519

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=i,d^2=f,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=eh,g^b=gh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286261, #72286394, #72286420
maximal normal subgroup of	#72286261, #72286394, #72286420
maximal quotient of	#72287519

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402715);