#506517

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2184	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

905 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319085, #72319586, #72319654
maximal normal subgroup of	#72319085, #72319586, #72319654
maximal quotient of	#72322973

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413713);