#7551123

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×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2267 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)(7,11)(8,9)(10,12)(13,14)(15,24)(16,21)(17,27)(18,28)(19,23)(20,26)(22,25);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,26,19,17,16,25)(14,20,21,24,23,22,28,27);(6,9)(10,11)(13,16,19,18)(14,21,23,28)(15,25,17,26)(20,24,22,27);(2,4)(5,7,8,12)(6,10,9,11)(15,17)(20,22)(24,27)(25,26);(13,17,18,25,19,15,16,26)(14,22,21,27,23,20,28,24)

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=g,e^2=f,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^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307384, #72307649, #72308409
maximal normal subgroup of	#72307384, #72307649, #72308409

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7458319);