#7564796

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2119 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307674, #72307710, #72308285
maximal normal subgroup of	#72307674, #72307710, #72308285

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7471992);