#431030

order	512
nilpotency class	3
isoclinism	isoclinic class of #417508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1614 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)(5,6,9,11)(7,12,8,10)(13,14,17,20)(15,18,22,27)(16,23,25,19)(21,28,24,26);(13,15)(14,18)(16,24,25,21)(17,22)(19,26,23,28)(20,27);(5,7,9,8)(6,10,11,12)(13,16,17,25)(14,19,20,23)(15,21,22,24)(18,26,27,28);(2,4)(5,8,9,7)(6,10,11,12)(13,15)(14,18)(16,21)(17,22)(19,26)(20,27)(23,28)(24,25)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338226);