#509906

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

614 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343483, #72343820, #72343843
maximal normal subgroup of	#72343483, #72343820, #72343843

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417102);