#184055

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

924 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)(3,4)(6,7)(11,12)(13,14,17,21,18,16,20,25)(15,19,24,27,22,23,26,28);(2,4)(5,6,9,11,8,7,10,12)(13,15)(14,19)(16,23)(17,24)(18,22)(20,26)(21,27)(25,28);(5,7,9,12,8,6,10,11)(15,22)(19,23)(24,26)(27,28);(13,16,17,25,18,14,20,21)(15,23,24,28,22,19,26,27)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,f^2=h,g^2=i,b^a=be,c^a=ci,c^b=ch

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91251);