#6447194

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	#34935	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2372 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)(7,10)(8,12)(9,11)(13,14,18,22,19,23,17,27)(15,24,25,21,26,28,16,20);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,26)(14,20,23,21)(16,17,25,18)(22,28,27,24);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,15,19,25,17,26)(14,21,22,28,23,20,27,24);(2,4)(13,16,18,15,19,25,17,26)(14,20,22,24,23,21,27,28);(13,17,19,18)(14,22,23,27)(15,25,26,16)(20,28,21,24)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fgi,d^2=f,e^2=i,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	#72308648, #72308933
maximal normal subgroup of	#72308648, #72308933

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354390);