#229064

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

790 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)(5,6,8,11,9,7,10,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,25)(24,26)(27,28);(2,3)(5,7,8,12,9,6,10,11)(13,15,16,23)(14,18,20,22)(17,24,25,28)(19,27,21,26);(14,19)(15,23)(17,25)(18,26)(20,21)(22,27);(1,3,4,2)(15,23)(18,22)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de,f^c=fh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318905, #72319475, #72319522, …
maximal normal subgroup of	#72318905, #72319475, #72319522, …
maximal quotient of	#72322667

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136260);