#6405604

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2410	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1866 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=df,e^c=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282258, #72282403
maximal normal subgroup of	#72282258, #72282403

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6312800);