#6465999

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2292 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321749, #72321844, #72321852
maximal normal subgroup of	#72321749, #72321844, #72321852

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373195);