#7503117

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2937 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)(7,11)(8,10)(9,12)(13,14)(15,20)(16,24)(17,19)(18,21)(22,28)(23,26)(25,27);(5,7)(6,9)(8,12)(10,11)(14,19)(15,22)(16,18)(20,27)(23,25)(26,28);(2,4)(13,15)(14,20)(16,22)(17,23)(18,25)(19,26)(21,27)(24,28);(2,4)(6,10)(7,12)(13,16,17,18)(14,21,19,24)(15,22,23,25)(20,27,26,28);(14,19)(20,26)(21,24)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308185, #72308252, #72308296
maximal normal subgroup of	#72308185, #72308252, #72308296

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410313);