#7985172

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284598, #72284682, #72284790, …
maximal normal subgroup of	#72284598, #72284682, #72284790, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892368);