#7966441

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

8372 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)(7,11)(8,9)(10,12)(13,14,19,23)(15,17,25,20)(16,26,28,21)(18,22,24,27);(1,2)(3,4)(6,9)(10,11)(13,15,19,25)(14,20,23,17)(16,18,28,24)(21,22,26,27);(5,6)(7,10)(8,9)(11,12)(13,16,19,28)(14,21,23,26)(15,24,25,18)(17,22,20,27);(5,7)(6,10)(8,12)(9,11)(13,16)(14,21)(15,18)(17,27)(19,28)(20,22)(23,26)(24,25);(5,7)(6,11)(8,12)(9,10)(13,17)(14,15)(16,27)(18,21)(19,20)(22,28)(23,25)(24,26);(5,6)(7,10)(8,9)(11,12)(13,18)(14,22)(15,16)(17,26)(19,24)(20,21)(23,27)(25,28)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304527, #72304577, #72304609
maximal normal subgroup of	#72304527, #72304577, #72304609

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7873637);