#7708765

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	#36624	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7028 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)(7,11)(10,12)(13,14,18,22)(15,23,24,19)(16,20,21,17)(25,26,27,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(15,24)(19,23)(25,27)(26,28);(6,9)(10,12)(13,15,18,24)(14,19,22,23)(16,27,21,25)(17,28,20,26);(13,16,18,21)(14,20,22,17)(15,25,24,27)(19,28,23,26);(5,6)(7,10)(8,9)(11,12)(13,17,18,20)(14,21,22,16)(15,26,24,28)(19,27,23,25);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,24)(25,28)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7615961);