#25808

order	256
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂×#577	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,11,17)(7,18,8,13)(9,24,26,32)(10,16,22,15)(12,21,28,20)(14,30,33,25)(19,34,23,27)(29,35,31,36);(5,7,10,20,11,8,22,21)(6,12,15,18,17,28,16,13)(9,19,25,35,26,23,30,36)(14,27,24,31,33,34,32,29);(1,2,3,4)(5,8)(6,13)(7,11)(9,19)(10,21)(12,15)(14,31)(16,28)(17,18)(20,22)(23,26)(24,34)(25,35)(27,32)(29,33)(30,36);(5,9,10,25,11,26,22,30)(6,14,15,24,17,33,16,32)(7,19,20,35,8,23,21,36)(12,27,18,31,28,34,13,29)

Polycyclic presentation:

2^8|a^2=h,b^2=e,c^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,d^c=dh,g^c=gh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#25808, #6443285, #6443286, …
maximal normal subgroup of	C₂×#25808, #6443285, #6443286, …
maximal quotient of	#299301, #299305, #299307, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 18796);