#7565801

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#44927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2232 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgi,d^2=fg,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=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308261, #72308510
maximal normal subgroup of	#72308261, #72308510

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7472997);