#6445117

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	#25125	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2063 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)(13,14,19,24)(15,20,26,16)(17,22,21,28)(18,27,25,23);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,24)(17,18)(19,26)(21,25)(22,27)(23,28);(2,4)(13,16,25,28,19,20,18,22)(14,21,27,26,24,17,23,15);(2,4)(13,17)(14,22)(15,25)(16,27)(18,26)(19,21)(20,23)(24,28);(5,7,8,11)(6,10,9,12)(15,26)(16,20)(17,21)(22,28)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6352313);