#380798

order	512
nilpotency class	3
isoclinism	isoclinic class of #29748
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1150 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)(7,10)(8,9)(11,12)(13,14,18,21)(15,22,24,19)(16,17,20,26)(23,25,27,28);(6,9)(10,12)(13,15,17,23,18,24,26,27)(14,19,20,28,21,22,16,25);(5,7)(6,10)(8,11)(9,12)(13,16,17,21,18,20,26,14)(15,22,23,28,24,19,27,25);(7,11)(10,12)(13,17,18,26)(14,20,21,16)(15,23,24,27)(19,28,22,25)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287994);