#7852187

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

8184 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)(13,14,20,26)(15,21,17,23)(16,24,18,22)(19,25,28,27);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15,20,17)(14,21,26,23)(16,27,18,25)(19,22,28,24);(5,7)(6,9)(8,12)(10,11)(13,16,20,18)(14,22,26,24)(15,25,17,27)(19,23,28,21);(13,17,20,15)(14,23,26,21)(16,25,18,27)(19,24,28,22);(5,7)(6,10)(8,12)(9,11)(13,18,20,16)(14,24,26,22)(15,25,17,27)(19,23,28,21);(5,6)(7,9)(8,11)(10,12)(13,19,20,28)(14,25,26,27)(15,22,17,24)(16,23,18,21)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=h,f^2=h,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	#72284609, #72284836, #72302985, …
maximal normal subgroup of	#72284609, #72284836, #72302985, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7759383);