#374479

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 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)(8,12)(9,10)(13,14)(15,20)(16,26)(17,22)(18,24)(19,23)(21,27)(25,28);(6,10)(11,12)(13,15,19,21)(14,20,23,27)(16,22,28,24)(17,25,18,26);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,27,19,28,17,20)(14,21,22,26,23,15,24,25);(5,8,9,7)(6,11,10,12)(13,17,19,18)(14,22,23,24)(15,25,21,26)(16,20,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281675);