#421898

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23391	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1311 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)(5,6,9,8)(7,11,12,10)(13,14,18,16,19,21,17,27)(15,20,24,26,25,28,23,22);(5,7)(6,10)(8,11)(9,12)(13,15,18,24,19,25,17,23)(14,20,16,26,21,28,27,22);(5,8,9,6)(7,11,12,10)(13,16)(14,18)(15,22)(17,21)(19,27)(20,23)(24,28)(25,26);(2,4)(7,12)(10,11)(13,17,19,18)(14,16,21,27)(15,23,25,24)(20,26,28,22)

Polycyclic presentation:

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

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, 329094);