#7728980

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

8134 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)(6,8)(10,11)(13,14)(15,22)(16,20)(17,19)(18,21)(23,28)(24,26)(25,27);(1,2,3,4)(13,15,18,17)(14,19,21,22)(16,23,25,24)(20,26,27,28);(5,6,9,8)(7,11,12,10)(15,17)(19,22)(23,24)(26,28);(5,7,9,12)(6,10,8,11)(13,16,18,25)(14,20,21,27)(15,23,17,24)(19,26,22,28);(5,8,9,6)(7,11,12,10)(13,17,18,15)(14,19,21,22)(16,24,25,23)(20,26,27,28);(13,14)(15,22)(16,20)(17,19)(18,21)(23,28)(24,26)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327640, #72330769
maximal normal subgroup of	#72327640, #72330769
maximal quotient of	#72327705, #72327953, #72328150, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7636176);