#456733

order	512
nilpotency class	3
isoclinism	isoclinic class of #456400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1032 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363929);