#388664

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

884 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)(8,11)(9,12)(13,14,17,21,18,16,27,20)(15,22,26,24,23,28,25,19);(1,2)(3,4)(5,7,10,6)(8,12,9,11)(15,23)(19,24)(22,28)(25,26);(2,4)(5,8,10,9)(6,11,7,12)(13,15,17,26,18,23,27,25)(14,19,21,22,16,24,20,28);(5,9,10,8)(6,11,7,12)(13,16)(14,18)(15,24)(17,20)(19,23)(21,27)(22,25)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295860);