#383336

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1228 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)(7,8)(9,11)(10,12)(13,14)(15,23)(16,19)(17,21)(18,22)(20,24)(25,27)(26,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,21)(17,19)(18,22)(23,24)(25,28)(26,27);(5,8,9,10)(6,7,11,12)(14,19)(16,17)(20,23)(21,22)(24,26)(27,28);(6,11)(7,12)(13,15)(14,20)(16,26)(17,24)(18,25)(19,23)(21,27)(22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290532);