#7679236

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

7141 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7586432);