#7714556

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

7806 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)(9,10)(11,12)(13,14)(15,20)(16,21)(17,19)(18,25)(22,26)(23,28)(24,27);(1,2,3,4)(13,15)(14,20)(16,19)(17,21)(18,23)(22,27)(24,26)(25,28);(6,10)(7,11)(14,17)(15,16)(22,25)(23,24);(5,7,9,11)(6,8,10,12)(13,16,19,15)(14,21,17,20)(18,24,26,23)(22,28,25,27);(5,8)(6,11)(7,10)(9,12)(13,17,19,14)(15,21,16,20)(18,25,26,22)(23,28,24,27);(13,18)(14,22)(15,23)(16,24)(17,25)(19,26)(20,27)(21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238207, #72238213, #72238253
maximal normal subgroup of	#72238207, #72238213, #72238253
maximal quotient of	#72256744, #72256765

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621752);