#6445169

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	#35525	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2077 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)(13,14,18,20)(15,17,22,19)(16,21,26,28)(23,27,24,25);(1,2)(3,4)(6,9)(10,12)(14,20)(15,22)(21,28)(23,24);(2,4)(13,15,19,14,18,22,17,20)(16,24,25,28,26,23,27,21);(2,4)(13,16)(14,21)(15,23)(17,27)(18,26)(19,25)(20,28)(22,24);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,20,15)(16,25,26,27)(21,23,28,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352365);