#7904591

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

7828 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)(5,6,9,11)(7,8,12,10)(13,14,18,17)(15,21,23,19)(16,24,25,20)(22,26,28,27);(1,2)(3,4)(13,15)(14,19)(16,22)(17,21)(18,23)(20,26)(24,27)(25,28);(5,7,9,12)(6,10,11,8)(13,15,18,23)(14,19,17,21)(16,22,25,28)(20,26,24,27);(5,7)(6,8)(9,12)(10,11)(13,15)(14,19)(16,22)(17,21)(18,23)(20,26)(24,27)(25,28);(5,8)(6,7)(9,10)(11,12)(13,16,18,25)(14,20,17,24)(15,22,23,28)(19,26,21,27);(13,17,18,14)(15,21,23,19)(16,20,25,24)(22,26,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304329, C₂×#7904591
maximal normal subgroup of	#72304329, C₂×#7904591
maximal quotient of	C₂×#7904591

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7811787);