#6463172

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	#35165	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2636 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370368);