#7988219

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

7357 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)(5,6,10,12)(7,9,11,8)(15,22)(20,26)(23,28)(24,27);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,14)(15,20)(16,21)(17,25)(18,19)(22,26)(23,27)(24,28);(5,7,10,11)(6,8,12,9)(14,19)(16,25)(20,26)(23,28);(5,8,10,9)(6,11,12,7)(13,15)(14,20)(16,23)(17,24)(18,22)(19,26)(21,27)(25,28);(5,7)(6,8)(9,12)(10,11)(13,16,18,25)(14,21,19,17)(15,23,22,28)(20,27,26,24);(5,9,10,8)(6,11,12,7)(13,17,18,21)(14,16,19,25)(15,24,22,27)(20,23,26,28)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,e^2=hi,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	#72308135, #72308285, #72308489
maximal normal subgroup of	#72308135, #72308285, #72308489

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7895415);