#383696

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1671 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)(7,9)(8,11)(10,12)(13,14)(15,19)(16,25,26,20)(17,27,23,21)(18,22)(24,28);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15,18,24)(14,19,22,28)(16,23,26,17)(20,27,25,21);(2,4)(5,8,10,9)(6,11,12,7)(13,16)(14,20)(15,23)(17,24)(18,26)(19,27)(21,28)(22,25);(5,9,10,8)(6,11,12,7)(13,17,18,23)(14,21,22,27)(15,16,24,26)(19,20,28,25)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313958, #72313971, #72313993
maximal normal subgroup of	#72313958, #72313971, #72313993

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290892);