#393401

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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

972 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)(7,10)(8,12)(9,11)(13,14)(15,21)(16,24)(17,22)(18,19)(20,26)(23,25)(27,28);(1,3,4,2)(5,7,9,8)(6,10,11,12)(13,15,21,14,18,22,17,19)(16,23,27,20,26,28,25,24);(5,8,9,7)(6,10,11,12)(14,19)(15,22)(20,24)(23,28);(13,16,17,25,18,26,21,27)(14,20,15,23,19,24,22,28)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,c^b=ci,g^d=gi

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, 300597);