#6445182

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352378);