#7879288

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

8354 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)(7,12)(8,10)(9,11)(13,14)(15,17)(16,26)(18,21)(19,22)(20,25)(23,27)(24,28);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,23)(17,22)(18,24)(19,25)(21,27)(26,28);(5,6)(7,12)(8,10)(9,11)(13,16,19,18)(14,21,22,26)(15,23,25,24)(17,28,20,27);(13,17,19,20)(14,15,22,25)(16,27,18,28)(21,24,26,23);(5,6)(7,10)(8,12)(9,11)(13,15)(14,20)(16,23)(17,22)(18,24)(19,25)(21,27)(26,28);(5,8,9,7)(6,10,11,12)(13,18,19,16)(14,21,22,26)(15,24,25,23)(17,28,20,27)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256671, #72256677, #72256685
maximal normal subgroup of	#72256671, #72256677, #72256685

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7786484);