#225451

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dei,e^b=eh,e^d=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132647);