#7685082

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

8494 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)(6,10)(7,11)(13,14)(15,24)(16,21)(17,22)(18,23)(19,20)(25,27)(26,28);(5,6,9,10)(7,8,11,12)(13,15,19,23)(14,18,20,24)(16,17,25,28)(21,26,27,22);(1,2,3,4)(5,7)(6,8)(9,11)(10,12)(14,20)(15,23)(16,25)(22,26);(1,2)(3,4)(5,8)(6,11)(7,10)(9,12)(13,16,19,25)(14,21,20,27)(15,17,23,28)(18,26,24,22);(13,17)(14,22)(15,25)(16,23)(18,21)(19,28)(20,26)(24,27);(13,18)(14,23)(15,20)(16,26)(17,27)(19,24)(21,28)(22,25)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304543, #72304584, #72304597
maximal normal subgroup of	#72304543, #72304584, #72304597

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592278);