#7633172

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×D₄×#41	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8390 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)(9,12)(13,14,18,22)(15,20,19,16)(17,21,26,28)(23,27,25,24);(1,2,3,4)(5,6,8,10)(7,9,11,12)(13,15)(14,16)(17,25)(18,19)(20,22)(21,27)(23,26)(24,28);(5,7,8,11)(6,9,10,12)(13,16)(14,19)(15,22)(17,24)(18,20)(21,25)(23,28)(26,27);(13,16)(14,15)(17,24)(18,20)(19,22)(21,23)(25,28)(26,27);(13,15)(14,20)(16,22)(17,23)(18,19)(21,27)(24,28)(25,26);(13,17,18,26)(14,21,22,28)(15,23,19,25)(16,24,20,27)

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540368);