#6452926

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3240 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)(7,10)(8,11)(9,12)(13,14,18,22)(15,20,24,27)(16,21,17,26)(19,25,28,23);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,26)(21,27)(22,28);(2,4)(5,7,9,8)(6,11,12,10)(13,15)(14,20)(16,25)(17,23)(18,24)(19,26)(21,28)(22,27);(2,4)(5,8,9,7)(6,11,12,10)(13,15)(14,20)(16,25)(17,23)(18,24)(19,26)(21,28)(22,27);(13,16,18,17)(14,21,22,26)(15,23,24,25)(19,27,28,20)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6360122);