#7501749

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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2641 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)(7,9)(8,10)(11,12)(15,22)(16,17)(19,26)(20,24)(23,25)(27,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14,18,21)(15,19,23,28)(16,24,17,20)(22,26,25,27);(2,4)(5,8,9,12)(6,10,7,11)(13,15)(14,19)(16,22)(17,25)(18,23)(20,27)(21,28)(24,26);(6,7)(10,11)(13,15)(14,19)(16,22)(17,25)(18,23)(20,27)(21,28)(24,26);(13,16,18,17)(14,20,21,24)(15,22,23,25)(19,27,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312770, #72312880, #72312897
maximal normal subgroup of	#72312770, #72312880, #72312897

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408945);