#6440596

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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

3010 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)(13,14,20,23,19,15,18,24)(16,25,17,22,28,21,27,26);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,19,14)(16,21,28,25)(17,22,27,26)(18,24,20,23);(2,4)(6,10)(9,12)(13,16,19,28)(14,21,15,25)(17,18,27,20)(22,23,26,24);(2,4)(13,17)(14,22)(15,26)(16,18)(19,27)(20,28)(21,23)(24,25);(13,18,19,20)(14,23,15,24)(16,17,28,27)(21,22,25,26)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6347792);