#7561854

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518904
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×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2426 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)(8,12)(9,10)(13,14,18,22)(15,23,25,20)(16,19,21,17)(24,28,27,26);(1,2)(3,4)(6,10)(11,12)(13,15)(14,20)(16,26)(17,24)(18,25)(19,27)(21,28)(22,23);(5,7,9,8)(6,11,10,12)(13,16,19,22,18,21,17,14)(15,23,24,26,25,20,27,28);(2,4)(5,8,9,7)(6,12,10,11)(13,16,19,22,18,21,17,14)(15,23,24,26,25,20,27,28);(6,10)(11,12)(13,17,18,19)(14,21,22,16)(15,24,25,27)(20,28,23,26)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fgi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307401, #72307649, #72308310
maximal normal subgroup of	#72307401, #72307649, #72308310

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469050);