#6357954

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(3,9)(4,11)(6,12)(8,14)(10,15)(13,16)(17,18,23,26)(19,21,25,22)(20,28,30,27)(24,31,32,29);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,22,18,23,25,21,26)(20,27,29,24,30,28,31,32);(1,3,4,9,5,10,11,15)(2,6,8,12,7,13,14,16)(17,20,21,31,23,30,22,29)(18,24,19,27,26,32,25,28);(3,10)(6,13)(9,15)(12,16)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,27,32,28);(2,7)(6,13)(8,14)(12,16)(20,30)(24,32)(27,28)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,e^b=ehi,d^c=di,e^c=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306837, #72306851, #72307028, …
maximal normal subgroup of	#72306837, #72306851, #72307028, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265150);