#7712700

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

7446 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)(6,10)(8,11)(13,14)(15,20)(16,21)(17,19)(18,25)(22,26)(23,28)(24,27);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15,19,16)(14,20,17,21)(18,23,26,24)(22,27,25,28);(6,10)(7,12)(14,17)(15,16)(22,25)(23,24);(13,16,19,15)(14,21,17,20)(18,24,26,23)(22,28,25,27);(5,7,9,12)(6,11,10,8)(13,17,19,14)(15,21,16,20)(18,25,26,22)(23,28,24,27);(5,8,9,11)(6,12,10,7)(13,18,19,26)(14,22,17,25)(15,23,16,24)(20,27,21,28)

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=i,f^2=i,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.

Relationships

maximal subgroup of	#72236836, #72236855, #72238258
maximal normal subgroup of	#72236836, #72236855, #72238258

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619896);