#8038982

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

7698 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)(15,16)(17,25)(18,19)(20,23)(21,26)(22,27)(24,28);(1,2,3,4)(5,7,10,11)(6,8,9,12)(14,19)(16,20)(21,25)(24,27);(5,6,10,9)(7,12,11,8)(13,15,18,23)(14,20,19,16)(17,22,26,28)(21,27,25,24);(5,8)(6,11)(7,9)(10,12)(13,15)(14,16)(17,22)(18,23)(19,20)(21,24)(25,27)(26,28);(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,24)(18,20)(19,23)(21,22)(25,28)(26,27);(5,9,10,6)(7,12,11,8)(13,17,18,26)(14,21,19,25)(15,22,23,28)(16,24,20,27)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,f^2=h,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	#72236858, #72238147, #72238198
maximal normal subgroup of	#72236858, #72238147, #72238198

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7946178);