#406260

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#469	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307703, #72307883, #72307946
maximal normal subgroup of	#72307703, #72307883, #72307946

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313456);