#7501368

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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2433 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:

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

Polycyclic presentation:

2^9|c^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307408, #72307796, #72307839
maximal normal subgroup of	#72307408, #72307796, #72307839

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408564);