#6487546

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470077
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×D₄×#41	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4348 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi

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, 6394742);