#7597013

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2298 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=h,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504209);