#475908

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214436, #72215022, #72216148, …
maximal normal subgroup of	#72214436, #72215022, #72216148, …
maximal quotient of	#72214149, #72214933

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383104);