#6440619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	#35250	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347815);