#7056296

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2026 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)(13,14,19,24,20,25,18,23)(15,16,26,22,27,28,17,21);(5,6)(7,11)(8,10)(9,12)(13,15,20,27)(14,21,25,22)(16,24,28,23)(17,18,26,19);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,17)(15,24)(18,21)(19,22)(20,28)(23,27)(25,26);(7,12)(9,11)(13,17,19,15,20,26,18,27)(14,22,24,28,25,21,23,16);(6,10)(9,11)(13,18,20,19)(14,23,25,24)(15,26,27,17)(16,22,28,21)

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=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6963492);