#7056248

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2039 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)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,22,20,23,18,28)(15,24,25,16,26,27,17,21);(5,6)(7,11)(8,10)(9,12)(13,15,20,26)(14,21,23,16)(17,18,25,19)(22,27,28,24);(2,4)(5,7,8,12)(6,9,10,11)(13,16,20,21)(14,15,23,26)(17,22,25,28)(18,24,19,27);(7,12)(9,11)(13,17,19,15,20,25,18,26)(14,16,22,27,23,21,28,24);(6,10)(9,11)(13,18,20,19)(14,22,23,28)(15,17,26,25)(16,27,21,24)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6963444);