#284429

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=eh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 191625);