#8058872

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

7699 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238146, #72238165, #72238207
maximal normal subgroup of	#72238146, #72238165, #72238207

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7966068);