#7135054

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#44936	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1750 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=h,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257290, #72257326, #72264324
maximal normal subgroup of	#72257290, #72257326, #72264324

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7042250);