#6401021

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#496	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1871 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6308217);