#7504026

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2137 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi

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, 7411222);