#359000

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2242 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)(9,10,12,11)(13,14,18,22)(15,20,24,27)(16,21,17,26)(19,25,28,23);(5,6)(7,8)(13,15,18,24)(14,19,22,28)(16,23,17,25)(20,21,27,26);(2,4)(6,8)(13,15)(14,20)(16,25)(17,23)(18,24)(19,26)(21,28)(22,27);(9,11,12,10)(13,16,18,17)(14,21,22,26)(15,23,24,25)(19,27,28,20)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,e^b=ei,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266196);