#7838099

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

7443 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)(13,14,18,17)(15,20,23,19)(16,24,26,21)(22,27,25,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,20)(18,23)(21,27)(24,28)(25,26);(6,8)(10,12)(13,15,18,23)(14,19,17,20)(16,22,26,25)(21,27,24,28);(7,11)(10,12)(13,15)(14,20)(16,25)(17,19)(18,23)(21,27)(22,26)(24,28);(5,7)(6,10)(8,12)(9,11)(13,16,18,26)(14,21,17,24)(15,22,23,25)(19,27,20,28);(5,8,9,6)(7,12,11,10)(13,17,18,14)(15,19,23,20)(16,24,26,21)(22,27,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238155, #72238220, #72238232
maximal normal subgroup of	#72238155, #72238220, #72238232

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7745295);