#382566

order	512
nilpotency class	3
isoclinism	isoclinic class of #29748
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×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1938 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:

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

Polycyclic presentation:

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

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, 289762);