#354699

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1298 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)(5,6,7,8)(13,14,19,21,18,22,16,17)(15,23,24,28,26,20,27,25);(6,8)(9,10,11,12)(13,15)(14,20)(16,27)(17,28)(18,26)(19,24)(21,25)(22,23);(1,2,3,4)(13,16,18,19)(14,21,22,17)(15,24,26,27)(20,28,23,25);(13,17,16,22,18,21,19,14)(15,25,27,20,26,28,24,23)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261895);