#7515181

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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

2183 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|c^2=f,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,d^c=di,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307757, #72308094, #72308327
maximal normal subgroup of	#72307757, #72308094, #72308327

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7422377);