#7670727

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

8416 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)(5,6,8,10)(7,9,12,11)(13,14,18,15)(16,19,23,21)(17,22,25,20)(24,27,28,26);(5,6)(7,9)(8,10)(11,12)(13,14,18,15)(16,19,23,21)(17,20,25,22)(24,26,28,27);(2,3)(5,6)(7,11)(8,10)(9,12)(14,15)(16,23)(20,22)(24,28);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302253, #72302931, #72302952
maximal normal subgroup of	#72302253, #72302931, #72302952

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577923);