#7518324

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514060
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
prime radical	#24354	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2256 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)(8,9)(10,12)(13,14)(15,22)(16,20)(17,19)(18,25)(21,26)(23,27)(24,28);(1,2)(3,4)(6,9)(10,11)(13,15)(14,18)(16,25)(17,24)(19,27)(20,22)(21,28)(23,26);(14,19)(15,23)(17,26)(18,22)(20,21)(24,25);(2,4)(14,19)(15,24)(17,26)(20,21)(23,25)(27,28);(5,7,8,12)(6,10,9,11)(14,20)(18,22)(19,21)(27,28)

Polycyclic presentation:

2^9|e^2=fi,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=cgi,d^b=dh,g^b=gi,h^b=hi,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	#72307576, #72307804, #72308220
maximal normal subgroup of	#72307576, #72307804, #72308220

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7425520);