#6384151

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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×#550	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2167 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)(6,9)(10,12)(13,14,20,25)(15,21,26,22)(16,23,17,27)(18,24,19,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,19)(17,18)(20,26)(22,25)(23,28)(24,27);(5,7,8,11)(6,10,9,12)(13,16,20,17)(14,22,25,21)(15,18,26,19)(23,24,27,28);(6,9)(10,12)(13,17,20,16)(14,23,25,27)(15,18,26,19)(21,28,22,24);(1,3,4,2)(13,18,20,19)(14,24,25,28)(15,16,26,17)(21,23,22,27)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291347);