#7689887

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

8509 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)(6,9)(10,11)(13,14,19,25)(15,18,24,20)(16,22,27,21)(17,23,26,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,17)(18,25)(19,24)(21,23)(22,28)(26,27);(5,6)(7,11)(8,9)(10,12)(13,16,19,27)(14,21,25,22)(15,26,24,17)(18,23,20,28);(5,7)(6,10)(8,12)(9,11)(13,16)(14,22)(15,17)(18,23)(19,27)(20,28)(21,25)(24,26);(13,17)(14,23)(15,16)(18,22)(19,26)(20,21)(24,27)(25,28);(13,18)(14,24)(15,25)(16,23)(17,21)(19,20)(22,26)(27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^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	#72304537, #72304543, #72304549
maximal normal subgroup of	#72304537, #72304543, #72304549

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7597083);