#7631241

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
prime radical	#36614	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342302, #72342344, #72342348
maximal normal subgroup of	#72342302, #72342344, #72342348

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538437);