#378233

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 285429);