#7691592

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

8229 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)(5,6,9,11)(7,8,12,10)(13,14,19,16)(15,17,20,22)(18,21,27,24)(23,25,28,26);(5,7)(6,10)(8,11)(9,12)(13,15)(14,17)(16,22)(18,23)(19,20)(21,25)(24,26)(27,28);(2,4)(5,8,9,10)(6,7,11,12)(14,16)(15,20)(21,24)(23,28);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16,19,14)(15,22,20,17)(18,24,27,21)(23,26,28,25);(13,17,19,22)(14,20,16,15)(18,26,27,25)(21,23,24,28);(13,18,19,27)(14,21,16,24)(15,23,20,28)(17,25,22,26)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=i,e^2=i,f^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	#72304618, #72304621, #72304639
maximal normal subgroup of	#72304618, #72304621, #72304639

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7598788);