#7864993

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

7371 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)(5,6,10,12)(7,9,11,8)(13,14,19,21)(15,20,18,22)(16,17,25,28)(23,24,26,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,21,22)(16,23,25,26)(17,24,28,27);(5,8,10,9)(6,11,12,7)(14,21)(16,25)(20,22)(23,26);(5,7)(6,9)(8,12)(10,11)(13,16)(14,17)(15,23)(18,26)(19,25)(20,24)(21,28)(22,27);(5,7)(6,8)(9,12)(10,11)(13,17)(14,16)(15,24)(18,27)(19,28)(20,23)(21,25)(22,26);(5,9,10,8)(6,11,12,7)(13,18,19,15)(14,22,21,20)(16,26,25,23)(17,27,28,24)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,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	#72285285
maximal normal subgroup of	#72285285

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7772189);