#346397

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3506	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=hi,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,d^b=dh,e^b=eh,f^b=fi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253593);