#6456532

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438967
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	#34909	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2015 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)(13,14,19,27,18,21,17,16)(15,22,28,20,26,23,25,24);(1,2)(3,4)(13,15)(14,20)(16,23)(17,28)(18,26)(19,25)(21,24)(22,27);(2,4)(6,9)(10,12)(13,14,19,27,18,21,17,16)(15,23,28,24,26,22,25,20);(2,4)(13,16,17,21,18,27,19,14)(15,24,25,23,26,20,28,22);(5,7,8,11)(6,10,9,12)(14,21)(16,27)(20,24)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257498, #72257512, #72264275, …
maximal normal subgroup of	#72257498, #72257512, #72264275, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363728);