#506503

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

976 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)(13,14,16,22,27,26,28,25,19,24,17,23,18,15,20,21);(5,7,8,12)(6,9,10,11)(13,15,17,25,27,14,20,23,19,26,16,21,18,24,28,22);(2,4)(14,21)(15,22)(16,20)(17,28)(18,27)(23,26)(24,25);(6,10)(9,11)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319085, #72319591, #72319642, …
maximal normal subgroup of	#72319085, #72319591, #72319642, …
maximal quotient of	#72322973

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413699);