#6457636

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2122 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,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, 6364832);